Mathematics of Chaos

An introduction of the basic math behind Chaos and the Mandelbrot set. Simple, yet accurate.

Fractal geometry and the insights of the science of Chaos are based on Complex Numbers. Unlike all other numbers, such as the natural numbers one through nine for instance 1.2.3.4.5.6.7.8.9, the Complex Numbers do not exist on a horizontal number line. They exist only on an x-y coordinate time plane where regular numbers on the horizontal grid combine with so called “Imaginary Numbers” on the vertical grid. Imaginary Numbers are simply numbers where a negative times a negative creates a negative, not a positive, like is the rule with all other numbers. In other words, with imaginary numbers -2 times -2 = -4, not +4. The Complex Numbers when iterated – subject to constant feedback – produce Fractal Scaling as is shown by the Mandelbrot set:

z -> z^2 + c

where c = any complex number.

-> means iteration, the feedback process where the end result of the last calculation becomes the beginning constant of the next: z^2 + c becomes the z in the next repetition. Like life it is a dynamic equation, existing in time, not a static equation.

When iteration of a squaring process is applied to non-complex numbers the results are always known and predictable. For instance when any non-complex number greater than one is repeatedly squared, it quickly approaches infinity: 1.1 * 1.1 = 1.21 * 1.21 = 1.4641 * 1.4641 = 2.14358 and after ten iterations the number created is 2.43… * 10 which written out is 2,430,000,000,000,000,000,000,000,000,000,000,000,000,000. A number so large as to dwarf even the national debt. Mathematicians say of this size number that it is approaching infinity.

The same is true for any non-complex number which is less than one, but in reverse; it quickly goes to the infinitely small, the zero. For example with .9: .9.9=.81; .81.81=.6561; .6561.6561=.43046 and after only ten iterations it becomes 1.39…10 which written out is .0000000000000000000000000000000000000000000000139…, a very small number indeed.

With non-complex numbers, such as real, rational or natural numbers, the squaring iteration must always go to infinity unless the starting number is one. No matter how many times you square one it will still equal one. But just the slightest bit more or less than one and the iteration of squaring will attract it to the infinitely large or small. The same behavior holds true for complex numbers: numbers just outside of the circle z = 1 on the complex plane will jump off into the infinitely large, complex numbers just inside z = 1 will quickly square into zero.

But the magic comes by adding the constant c (a complex number) to the squaring process and starting from z at zero: z -> z^2 + c. Then stable iterations – a set attracted to neither the infinitely small or infinitely large – become possible. The potentially stable Complex numbers lie both outside and inside of the circle of z = 1; specifically on the complex plane they lie between -2.4 and .8 on the real number line, the horizontal x grid, and between -1.2 and +1.2 on the imaginary line, the vertical y grid. These complex numbers in effect stay within the meso-cosmic realm, the world of Man, even if the z -> z^2 + c iteration process goes on forever. These numbers are contained within the black of the Mandelbrot fractal.

In the Mandelbrot formula z -> z^2 + c, where you always start the iterative process with z equals zero, and c equaling any complex number, an endless series of seemingly random or chaotic numbers are produced. Like the weather, the stock market and other chaotic systems, negligible changes in quantities, coupled with feedback, can produce unexpected chaotic effects. The behavior of the complex numbers thus mirrors the behavior of the fourth dimension, the real world where Chaos is obvious or lurks behind the most ordered of systems.

With some values of c the iterative process immediately begins to exponentially increase or fall into infinity. These numbers are completely outside of the Mandelbrot set of “meso-cosmic” dynamics. With other values of c the iterative process is stable for a number of repetitions, and only later in the dynamic process are they attracted to infinity. These are the unstable strange attractor numbers just on the outside edge of the Mandelbrot set. They are shown on computer graphics with colors or shades of grey according to the number of stable iterations. The values of c which remain stable, repeating as a finite number forever, never attracted to infinity, and thus within the mesocosmic set, the Mandelbrot set, are plotted as black.

Some iterations of complex numbers like 1 -1i run off into infinity from the start, just like all of the real numbers. Other complex numbers are always stable like -1 +0i. Other complex numbers stay stable for many iterations, and then only further into the process do they unpredictably begin to start to increase or decrease exponentially (for example, .37 +4i stays stable for 12 iterations). These are the numbers on the edge of inclusion of the stable numbers shown in black. Chaos enters into the iteration because out of the potentially infinite number of complex numbers in the window of -2.4 to .8 along the horizontal real number axis, and -1.2 to 1.2 along the vertical imaginary number axis, there are an infinite subset on the edge which are subject to the unpredictable strange attractor. All that we know about these edge numbers is that if the z produced by any iteration lies outside of a circle with a radius of 2 on the complex plane, then the subsequent z values will go to infinity, and there is no need to continue the process.

By using a computer you can escape the normal limitations of human time. You can try a very large number of different complex numbers and iterate them to see what kind they may be. Under the Mandelbrot formula you start with z equals zero and then try different values for c. When a particular value of c is attracted to infinity – produces a value for z greater than 2 – then you stop that iteration, go back to z equals zero again, and try another c, and so on, over and over again, millions and millions of times as only a computer can do.

Mandelbrot was the first to discover that by using zero as the base z for each iteration, and trying a large number of the possible complex numbers with a computer on a trial and error basis, that he could define the set of stable complex numbers graphically by plotting their location on the complex plane. This is exactly what the Mandelbrot figure is. Along with this discovery came the surprise realization of the beauty and fractal recursive nature of these numbers when displayed graphically.

The black parts of the Mandelbrot fractal plot the stable iterations on the complex plane. When a complex number is attracted to infinity, small or large, it is either not plotted on the graph or is shown as a color according to the number of iterations it takes before the complex number begins its exponential spiral into infinity.

Every point in the plane of complex numbers is either outside the Mandelbrot set, infinite, or inside of it, finite. The Mandelbrot fractal thus portrays two-dimensionally the infinity between the whole numbers zero and one, the potential and the actual. This is the world of Man and the basis of all computer operations. The border which defines our area between the finite and the infinite – where our potential can come into actuality – is impossible to determine exactly. It is subject to the strange attractor. You never know when you may fall into or out of it, or how. The closer you look, the more you magnify by choosing a new c close to the last one chosen, the more the fractal complexities repeat with recognizable patterns – but rarely identical – to define an infinitely irregular border.

Only by plotting these numbers in time using an iterative process and two- dimensional representation is the hidden order and great beauty of the fourth dimensional complex numbers revealed. The infinitely recursive nature of the Mandelbrot fractal is the truly astonishing feature of the Mandelbrot and other fractal sets. Again, infinitely recursive means that the basic shapes of the overall form repeat themselves, but with variations, no matter how close you look at the detail. There is self similarity or self affinity from one scale to the next. As you magnify and look deeper and deeper into the microcosm of the figure, you find the same basic forms are repeated, but are still different and unique. At each scale the fractal is viewed there is a consonance of similarity with the original form, a repeating self similarity. This kind of fractal recursiveness seems to be a basic Law for all of reality, even consciousness. This may explain the feeling of “deja vu” which we all experience from time to time.

The primary shape of the Mandelbrot fractal are the two black blobs or warts, called “atoms” by Mandelbrot. The large kind of heart shaped black blob on the right is called a “cardioid,” and the smaller black figure on its left is “disk” like. Both the cardioid and the disk each have an infinity of smaller black disk like shapes surrounding them, and each of these smaller black disks in turn has an infinity of similar and still smaller black disks around it, and so forth ad infinitum.

To the left of the large atom, extending from a line to the left of the large sphere you will find another smaller cardioid; magnifying you will see more and more cardioids radiating out all over the large atom, and out again from each of the smaller atoms, and so forth, again to infinity. The black atoms, which plot the complex numbers within the stable set, are infinitely recursive, or self similar.

So too are the colored shapes next to the black atoms. The geometric shapes repeat with slight variations in various sizes approaching the infinitely small as the details of the edge of the set are magnified. For a small example of this jump to my Mandelbrot magnification animation which I call “spidey.” There are many other sites on the web where you can see this magnification for yourself in far greater detail. Try an Internet search engine of the word fractal to find the latest sites. There are hundreds of different animations and thousands of images available on the Web to illustrate these concepts. Many allow you to magnify and zoom in on any point of the Mandelbrot Set. Your in control and can explore as you like. The images you can create there will beautifully illustrate the Mandelbrot set and other fractals. Only by doing and seeing this for yourself can you fully understand fractal recursiveness over scales of magnitude. Left brain words alone don’t do it justice. Here is a little sample to give you a taste.

The Mandelbrot set is holistic and continuous. It has been proven mathematically that all of the black atoms of the Mandelbrot fractal are touching, connected, even though sometimes only through extremely thin lines or filaments that require billions of scales of magnification to become visible. Thus, for instance, all of the extremely small Mandelbrots shown in next picture are connected with the big Mandelbrot by a black line which is too small to be visible in the picture. Higher levels of magnification would reveal the connections. There are an infinite number of Mandelbrot figures embedded in the overall geometric figure. They exist on different scales of magnitude and start to become visible with magnification. On each of these Mandelbrot figures themselves there are an infinite number of black cardioids or atoms. The smaller Mandelbrots and other geometric figures radiate outward from the large mandelbrot. This is shown in the following magnified image of the left side of the figure:

The Mandelbrot set has continuity and cohesive identity. For this reason it is a perfect symbol for the collective unconscious and the full integration of consciousness in being, in the black of Awareness. There is tremendous diversity in the Mandelbrot set within its cohesiveness. This is what makes its continuity and identity all the more remarkable, and allows it to serve so well as a world model of consciousness. Some small flavor of the great variety of geometric shapes contained with the Mandelbrot set is shown by the following magnifications of a few of the many territories in this vast world. The coloring of the first picture depicts the Mandelbrot set as a whole to show how lines of force, lines of order and cosmos, radiate out from the Mandelbrot figure into the chaos. Other sets of complex numbers produce sets of static complex numbers and fractals by using the same iterative formula z -> z^2 + c but they do not start with z = 0 after the complex vector falls into infinity and try a new value for c. Instead they keep the same value for c and use a new value for the beginning z. They are called Julia sets after the French mathematician Gaston Julia who was the first to study iteration with complex numbers in the 1920’s. A segment of a spiral shaped Julia set is shown below.

Julia was able to visualize the fractal recursive nature of these numbers, even without the aid of computer graphics. He did so by a solid year of contemplation while lying in bed recuperating from a serious injury suffered in World War One. Unlike the Mandelbrot set which samples all values of c to test whether they are attracted to infinity or not, the Julia sets are based on a fixed value for c and the value of the beginning z less than 2 is varied over time. There are an infinite number of different Julia sets possible. But unlike the Mandelbrot set grounded in zero where the black portions are all connected with each other in the Complex Plane, the different Julia sets are disconnected with each other. Each Julia set stands alone, segregated from the other Julias. The Julia sets whose values of c lie within the Mandelbrot set are internally connected and cohesive, even though externally disconnected with each other. The Julias sets whose values of c lie outside of the Mandelbrot set are also internally disconnected, falling apart like Cantor’s dust, and so are called Cantor sets. Cantor type Julia sets are good symbols for average fragmented consciousness before individuation. The externally separate, but internally connected Julia sets are beautiful symbols of partially integrated individuality, striving Man, not yet fully one with the collective consciousness, not yet grounded in the zero of the Mandelbrot set.

Even without the advantage of computer plotting Julia and a few other mathematicians in the 1920’s knew that iteration of complex numbers produced fractals with recursive features, but they did not comprehend the full significance of the process, nor did they think to stabilize the dynamics in zero. Mandelbrot was the first to realize that this was the geometry of nature, the reality of the fourth dimension, and not just some meaningless bizarre fluke of mathematics. His discovery was based on ZERO tying all of the finite complex numbers together by grounding the z in zero and floating the c. The parallel with tying all of the states of consciousness together with Awareness in the integration process is profound and exciting. This insight appears to have been first made by Arnold Keyserling at the time the first School of Wisdom was opened in America in 1992. The philosophic implications of the Mandelbrot set are far reaching. A basic Law of Wisdom has been discovered by Mandelbrot, although, unlike Keyserling, he appears to be uninterested or unaware of its non-mathematical implications.

Mathematically, Mandelbrot discovered that his holistic fractal governs and defines all of the Julia sets. Julia sets whose value of c lie within the set of the Mandelbrot fractal, within the black atoms, are internally connected, holistic. But Julia sets whose value of c lies outside of the Mandelbrot fractal on the plane of complex numbers, the Cantor Sets, are fragmented into infinitely many pieces. The further from the black edge, the quicker the Julia sets break up and fall into dust. The Julia sets with a value of c just on the outside of the black border, the edge of the Mandelbrot set, are the most complex and beautifully ornate of all.